Question

A 4.00 kg rock hangs from a 2.00 kg chain which is 1.50 m long.

(a) Find the tension in the chain as a function of the distance from the bottom.

(b) How much time does it take a pulse to travel from the bottom of the chain to the top of the chain? (Hint: The tension in the chain is NOT constant.)

(c) As the pulse travels up the chain, does it become wider or narrower? Explain using grammatically correct sentences and a diagram or two

Answer 1

given

mass of rock, M =4 kg

mass of chain, m = 2 kg

l = 1.5 m

a. at distnace x from the bottom

tension (T(x)) = Mg + (m/l)*xg = g(M + mx/l) = g(4 + 2x/1.5)

b. consider distance dx at distance x form the bottom of the chain

speed at this position v = sqrt(Tl/m)

now v = dx/dt

dt = dx/v = dx*sqrt(m/gl(4 + 2x/1.5))

integrating

t = sqrt(m/gl)[6*sqrt(5*1.5/3 + 4)/5 - 6*sqrt(4)/5] = 0.24310367189 s

c. as pulse travels up, T increases, hence v increases

now, assuming constant frequency,

f*lambda= v

v increases, hence lambda increases

hence the pulse becomes wider as it goes upwards